Finding a Degree Function: Let $R=\Bbb Z[i]$ and let $d(x)=N(x)$ for $x∈R$, where $N(x)$ is the norm of $x$

Question

(a) For $x,y∈R$−{0},show that $d(x)≤d(xy)$

(b) Let $x=2+10i$ and $y=5+3i$. Find $q,r∈R$ such that $x=yq+r$ where either $r=0$ or $d(r) < d(y)$

For the first part of the question I don't know how I would prove this but it seems obvious that $x≤xy$ so I am not sure logically how I would say this.

For the second question I have so far $5+3i/2+10i$=$(5+3i)(2-10i)/(2+10i)(2-10i)$=$40-44i/104$ but I don't know how I would proceed after that to show what it is asking...

Answer 1

For $(a)$, use the fact that $d$ is a multiplicative function: $d(xy)=d(x)d(y)$.

For $(b)$, you can write $$2+10i = (a+bi)(5+3i) + (c+di).$$

Simplify to get $$2+10i = 5a - 3b + c + (3a+5b+d)i.$$

Now see if you can spot a solution $(a, b)$ such that $c^2 + d^2 < 5^2 + 3^2 = 34$.